Stochastic resonance analysis of a coupled high-speed maglev vehicle-bridge coupled system under bounded noise

Coupled oscillations typically occur in maglev vehicle-bridge coupled systems excited by bounded noise caused by guideway irregularities. The paper employed Hamilton equations to derive the corresponding canonical transformation equations and determined the critical stable regions for two kinds of resonances using the largest Lyapunov exponents. The results show that the critical stable region between the excitation amplitude and the resonant frequency ratio is a valley shape when the system has external resonance only. When considering both internal and external resonances, the critical stable region between the excitation amplitude and resonant frequency ratio presents a small saddle shape. Energy transfers from the first to the second oscillator under with both internal and extrinsic resonance. As the guideway irregularities’ coefficients increase, the maximum Lyapunov exponents of the two conditions change from negative to positive, which means that the system varies from a stable state to instability.

www.nature.com/scientificreports/ resonance in a bistable electronic device: a tunnel diode 17 . Stochastic resonance was detected using a simple experimental setup by investigating the time evolution of the voltage measured across the tunnel diode as a function of the input noise intensity. (ii) the stochastic resonant phenomena was studied experimentally and theoretically for a state-of-the-art metal-oxide memristive device based on yttria-stabilized zirconium dioxide and tantalum pent-oxide, which has exhibited bipolar filamentary resistive switching of the anionic type 18 . The optimal noise intensity corresponding to the stochastic resonance phenomenon was interpreted using a stochastic memristor model by adding an external noise source to the control voltage. Furthermore, dynamical systems have been studied in the presence of relevant noise-induced phenomena with a constructive role in stability, such as, (i) the damping-enhanced stability of a Brownian particle starting from an unstable initial position and moving in a metastable system was explored 19 ; (ii) a memristor used for resistive switches behaved as multistable nonlinear systems between low-resistance and high-resistance states in a random telegraphic signal mode 20 ; (iii) an approach using a real stable polynomial combined with a Gauge transformation was presented and the bistability of polynomials corresponding to factors of the original multi-graph model resulted in real stable polynomials of each factor in various multi-graph models of the aforementioned contraction sequence 21 ; (iv) a nonstationary function within a memristive system was investigated to devise a simplified description of transient processes under different noise intensities, and the relaxation time was obtained, which depended nonmonotonically on the intensity of the fluctuations 22 . Finally, in other scientific study fields, such as quantum phase transitions in complex biological and physical systems, the positive role of noise has also been demonstrated. For example: (i) the Gaussian non equilibrium steady states of the quantum characteristics of such critical phenomena have been reviewed 23 .
(ii) a quantum case has been detected in which the indeterminacy arising from the uncertainty principle reduced the accuracy of the parameter estimation in a way that cannot be neglected, even in the limit of infinite copies 24 .
(iii) the phenomena of dissonance and consonance have considered, where two sensory neurons were driven by noise and subthreshold periodic signals, and their outputs plus noise were applied to a third neuron with noise added to them 25 ; (iv) the noise in the high resistive state was found to be featured by nearly the same probability density functions and spectrum as the inner noise of the experimental setup 26 . Compared with previous research frameworks of time-domain samples in Newtonian mechanics, a Hamiltonian system with narrow-band random excitation is more complex. Some theoretical bases have been proposed. Colored noise refers to a fixed centre frequency, white noise intensity, and a uniform distribution angle with a triangular relationship 27,28 , which examined the responses. This noise was utilized in a Wiener process by an equivalent to measured power spectra in methods put forward by Chen Zeshen, Jin Zhibin, and Jin  , who performed the theoretical modeling analysis in the time domain with covariance analysis method, generated guideway irregularities by combining the shape filter with the time delay system, and considered short-wavelength track irregularities. Some examples substantiated in bridge responses under wind loads have been obtained by Dimentberg M, Lin Y.K., and Jian Deng. [32][33][34] , who obtained the subcritical responses to an external broadband random, considered turbulence stabilize even a single-degree of freedom structural motion, and provided insights on how to analyze and control parametric resonances under a bounded noise process in engineering applications. Since Lyon et al. first applied a stochastic averaging method 35 proposed by R. L. Stratonovich 36 , it was subsequently applied by Zhu W. Q., Huang Z. L., Liu Zhonghua, and W.Y. Liu 37,38 , who proposed a stochastic averaging method to predict approximately the response of quasi-integrable Hamiltonian systems excited by bounded noise, determined the threshold of bounded noise amplitude for the onset of chaos. They have applied to duffer oscillator analyses using the random mean principle and the limited differential technique. Although Bo Zhang 39 investigated the random stability of a suspended wheelset system considering Gaussian white noise by the random average method. At present, there are few studies on resonance based on stochastic stability. Solving the resonant behaviour of the complex maglev vehicle-bridge coupled system is key to the further development of EMS.
The study presented in this paper aims to build a model to explore the critical conditions of stochastic resonance over the whole bridge span with the aerodynamic loads and guideway irregularities. Hamilton's theory is applied to derive the differential equations and their dimensionless equations 28 . The appropriate stable domains at different resonances based on stochastic averaging theory and canonical transformations are given. The stability probability according to the Fokker-Planck-Kolmogorov (FPK) equation utilized in this study. Moreover, a unique numerical method for assessing the effects of aerodynamic loads and the guideway excitation on the stochastic resonance of the maglev vehicle-bridge coupled system is also presented.

The stochastic averaging method
To explain the theoretical basis of our model's analysis, the derivation process of the stochastic averaging method is introduced below.
Consider a quasi-integrable Hamiltonian system under bounded noise excitation governed by the following equations of motion 37 : where Q i and P i are the generalized displacements and momentum, respectively; H = H(Q, P) is the Hamiltonian; εc ij = εc ij (Q, P) are the coefficients of lightly linear or nonlinear damping; εh ij = εh ij (Q, P) denotes the amplitudes of weak bounded noises; and ξ k (t) represents independent bounded noises of the form where k and σ 2 k are constants representing the center frequencies and strengths of the frequency perturbations, respectively; B k (t) are independent units in the Wiener processes; and k are independent random phases that www.nature.com/scientificreports/ are uniformly distributed in [0, 2π]. ξ k (t)are independent stationary random processes in a wide sense with spectral densities and auto correlation functions The bandwidths of the processes ξ k (t) depend mainly on the parameters σ k . The processes are narrow-banded when σ k are small and wide-banded processes when σ k are large. It is assumed that σ k are small and thus the corresponding processes are narrow-band.
Suppose that the Hamiltonian system shown with Eq. (1) with ε = 0 is integrable, i.e., there exists a set of canonical transformations through which new Hamiltonian equations are of the following form: where I i and ω i are action variables and frequencies, respectively; θ i are the angle variables conjugated to I i ; and H(I) is the transformed Hamiltonian, which is independent of θ i . The Hamiltonian system is resonant if there exist α(1 ≤ α ≤ n − 1) resonant relations such that where L u i are integers that are not all zero for a fixed u. By using the canonical transformations of Eq. (5), the differential equations for the action and angle variables of the quasi-integrable Hamiltonian system (1) can be obtained from Eq. (1) as follows: The form and dimension of the averaging equations depend on the resonance of the Hamiltonian system described in Eq. (8). In the following subsections, two cases are considered.
External resonance only. Consider a system with external resonance but no internal resonance. Suppose that there are β external primary resonant relations between the first β oscillators and the first β bounded excitations, i.e., 27 where M v and L v are positive or negative integers and there is no summation over subscript υ. Introduce β new variables: Using the transformation in Eq. (10), the differential equations for I 1 , ..., I n, ψ 1 ...., ψ n , θ 1 ...., θ n can be obtained from Eq. (8) as follows: www.nature.com/scientificreports/ where I = (I 1 , ..., I n ), ψ = (ψ 1 , ..., ψ β ),θ = (θ 1 , ..., θ n ) As shown in Eq. (11), I 1 , ..., I n , and ψ 1 , ..., ψ β are slowly-varying processes, while θ 1 , ..., θ n are rapidly varying processes. By applying deterministic averaging to θ 1 , . . . , θ n , the averaged ITȮ equations can be defined as follows: where The averaged FPK equation associated with Eq. (12) is where p = p(I, �, 0|I 0 , ψ 0 ) is the transition probability density. The initial condition of Eq. (14) is The boundary conditions with respect to ψ v are periodic, i.e., The boundary conditions with respect to I r are defined as The reduced FPK equation with its boundary conditions can be solved numerically by using the combination of finite difference method and the successive over-relaxation method.
Both internal and external resonance. The transformation is shown in Eq. (20), the differential equations for I, ψ, �, and θ 1 can be obtained from Eq. (8) as follows.

Motion model
To better introduce the case applications, we first provide a detailed description of the composition of the specific model and the parameter settings is provided.
Theoretical hypothesis. In general, model complexity is determined according to the purpose of the model. The model should be sufficiently comprehensive to allow the reliable and accurate analysis of vibration response analyses in terms of ride comfort and safety. Motion stability of an EMS model refers to the parameters for a minimal coupling model composed of a maglev vehicle and a bridge including the elite segments proposed by Jin-hui Li 6 . Depending on the basic elements analyzed, the complex systems is then simplified to minimum models, which is more efficient. Table 1 lists the correlation variables.
The fundamental assumptions are described as follows: www.nature.com/scientificreports/ • Electromagnet forces are linear.
• The system is decoupled both laterally and vertically without considering the turning radius, height difference or rolling freedom. • The random irregularity is bounded noise applied with the shaping filter technique.
• A Bernoulli-Euler beam is adopted for the calculation of the bridge model. • The moving mass and the action point of the concentrated force are at the geometric center of the electromagnets.
Modelling of substructures. Bridge model. Based on the above analysis, the minimal model is presented in Fig. 1 6, 10, 40 . The loads of vehicle and passengers are equivalent to a weight force acting on the center of the electromagnetic mass. The vehicle-bridge coupled system can be described using the structure is shown in Fig. 1.
The electromagnetic forces are uniformly distributed on the bridge and the electromagnet. The current or voltage of the magnet that controls the electromagnetic action is applied to adjust the gap between the electromagnet and the bridge. The bridge is also shown in Fig. 1, where the endpoint marked with "0" is taken as the coordinate origin. The direction of the hammer is the positive direction of y-axis. Considering the high stiffness of the electromagnet, its deformation in the y-direction can be ignored. The dynamic characteristics have a considerable influence on the elastic deformation of the bridge in the y direction. Based on the above assumptions, Fig. 1 illustrates a simplified suspension electromagnet-bridge coupling model. Where y B is the vertical displacement of the bridge, y E is the vertical displacement of the electromagnet relative to the reference plane, and δ is the distance between the electromagnet and the bridge.
The vertical motion of the bridge can be formulated as:  Bounded noise includes harmonic variations with a maximum amplitude, a constant frequency, and random phases [45][46][47][48][49][50] . It can be expressed via the stationary random process ξ 1 (t) as follows: where Ω is a constant center frequency with Ω = π V/L, with V being the vehicle speed and L being the bridge length, A v is the maximum deflection of the bridge in the vertical direction, t is time, σ 1 is the strength of the frequency perturbations, B(t) is a unit Wiener process, and Δ is a random phase that is uniformly distributed in [0, 2π] 37 . Its auto covariance functions are and their corresponding spectral densities are .. A comparison between the filter, the experimental line, and the literature is shown in Fig. 2, where it is evident that there is numerical consistency.

Dynamic differential equations.
In the above equations, σ is the amplification factor of multiple suspension units. Through simplification, the dynamic differential equations can be expressed as: .. .

Analysis of the Lyapunov exponent and stationary probability
To better evaluate the case study, the calculation process is elaborated further. The differential equations for the motion integrals H 1 and H 2 and the angle variables θ 1 and θ 2 are expressed as follows: where Only external resonant vibration. Consider a system with external resonance but no internal resonance.
Suppose that there is a single external resonant relation 36 .
where ε and Θ can be regarded as small detuning parameters. The new variable ψ is introduced, as defined in Eq. (39).
The differential equations for H 1 , H 2 , and ψ, are stated below: The differential equations for ρ and α 1 can be formulated as follows The averaging FPK equation presented by Zhu (2002) [17] that is associated with ψ is The solution that satisfies the periodic condition is where C is a normalized constant. The mean dα 1 can be calculated as: www.nature.com/scientificreports/ If A(α) > 0 , α 1 → 1 and p(α 1 , ψ) = p(ψ)δ(1) . if A(α) < 0 , α 1 → 0 and p(α 1 , ψ) = p(ψ)δ(0) . The equation can be expressed as: Some results we obtained via simulations. The joint probability density p(α 1, ψ) represents the centralized peak when Ψ = 0 and α 1 = 0. In Fig. 3, when ψ = 0 and α 1 = 0.5 , the stationary joint probability density p(α 1, ψ) shows a peak. In Fig. 3, when � 1 −2ω 1 = 0 , the first time of resonating to exceed is the shortest. As shown in Fig. 3 and Fig. 4, the cross-stable region in the frequency-excitation amplitude plane has a valley shape when 1 = 0 . As the guideway irregularity coefficient E 11 increases, the maximum Lyapunov exponents increase gradually from their initial small stable state, as shown in Fig. 5. A comparison between the stochastic averaging method and the numerical simulation is also shown in Fig. 5, where the numerical consistency between the results is evident. The random average method is more vivid from the grasp of the critical value of total energy and the changing trend. Through the grasp of displacement, the numerical simulation has a large amount of calculation.
Both internal and extrinsic resonance. Consider a case with primary external resonance between the first bounded noise excitation and the first oscillator. The primary internal resonance between the two oscillators [36] can be expressed as: where and η are detuning parameters 26 . The new variables ψ and are introduced as angle differences.
The maximum Lyapunov exponent can be expressed as: A numerical calculation is helpful for determining reason for this resonance. In Fig. 6, when ψ = 0, = 0, the stationary joint probability density p(Φ , ψ) shows a peak. The joint probability density p(Φ , ψ) represents a centralized distribution with the angle differences Ψ = 0 and Φ = 0. Figures 6 and 7 show the stable and unstable regions in the frequency-excitation amplitude plane, which is resembles a saddle shape. As the guideway irregularity coefficient E 11 increases, the maximum Lyapunov exponents start from their initial small stable state and rise in a step-wise manner, as shown in Fig. 8. A comparison between the stochastic averaging method and the numerical simulation is also shown in Fig. 8, where numerical consistency between the stochastic averaging method and the numerical simulation is also shown in Fig. 8, where numerical consistency can be observed. The random average method is more vivid from the grasp of the critical value of total energy and the changing trend. Through the grasp of displacement, the numerical simulation has a large amount of calculation. 1*10 -6 2*10 -6 3*10 -6 4*10 -6 5*10 -6 6*10 -6 7*10 -6 Figure 5. Lyapunov exponent in a system with external excitation only.

Summary
In this paper, stochastic averaging method for a quasi-integrable Hamiltonian system under bounded noise is proposed in this paper. The forms and dimensions of the averaging equations depend on the number of internal and external resonant relations in the system. The proposed procedures were applied in the prediction of a high-speed maglev train-bridge coupled system responses under bounded noise. The results obtained from the reduced averaging FPK equation by using the finite difference and the successive over-relaxation iterative methods are consistent with simulations of the original system. It is noted that the proposed procedure may also be applicable in studying the reliability and stochastic stability of these systems under bounded noise. The results conclusively show that • The joint probability density of different phases has a peak when the phases are close to each other.
• The stable region shrinks when the two resonance conditions are satisfied.
• When the unstable region in the phase diagram (E 11 , Ω / 2ω 1 ) is affected by only one external resonance, the external resonance reduces the stable region. The closer the external resonance frequency is to the system frequency, the smaller the size of the stable region. Moreover, as E 11 increases, the maximum Lyapunov exponent changes from negative to positive, and the system shifts from stability to instability in a nearly linear manner. • When the unstable region in the phase diagram (E 11 , Ω / 2ω 1 ) is affected by both internal and external resonance, the stable region shrinks as the energy is transferred from the first oscillator to the second oscillator during the two resonances. As E 11 increases, the maximum Lyapunov exponent changes from negative to positive, and the system shifts from stability to instability in a step-wise manner.  www.nature.com/scientificreports/

Data availability
The data used in this study can be obtained from the corresponding author upon request.
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